Counting the chromatic layers lost when taking fixed points

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Family 314 · Topology
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-22

1. The question

A symmetry can reveal topology that is invisible before taking fixed points. Classical Smith theory gives an important constraint: if a finite complex has trivial reduced homology modulo a prime $p$, then so does its fixed-point space under a finite $p$-group.

Chromatic homotopy theory asks a more finely graded question. Instead of testing all mod-$p$ homology at once, it uses theories $K(0),K(1),K(2),\ldots$, each detecting a different periodic layer. How many layers can be lost when passing from points fixed by a subgroup to points fixed by the whole group?

Family 314 claims an exact answer for every finite $p$-group and every embedded subgroup. The answer is a group-theoretic number: the fewest cyclic steps in a subnormal chain joining the subgroup to the group. Remarkably, the claimed number is independent of the chromatic height at which the question starts.

2. Background and essential definitions

A finite $p$-group has order a power of the prime $p$. If it acts on a space $X$, the fixed-point set $X^H$ consists of points unchanged by every element of a subgroup $H$.

The stable version uses genuine equivariant spectra, which remember suspension not only by ordinary spheres but also by spheres carrying linear group actions. Their geometric fixed points, written $\Phi^H X$, isolate the part associated with the subgroup $H$. On suspension spectra of appropriately based spaces, this construction recovers the corresponding fixed-point information. It is distinct from homotopy fixed points, which encode a different construction.

The theorem concerns finite $p$-local genuine spectra, including compact retracts. Here “finite” refers to a finite cell construction, and “$p$-local” means that all other primes are inverted.

Morava $K$-theory $K(i)$ detects chromatic height $i$, with $K(0)=H\mathbb Q$, rational homology. A spectrum is $K(i)$-acyclic when its $K(i)$-homology is zero. For a nonzero finite spectrum, the first nonvanishing height is its chromatic type.

For $H\leq G$, define $r_n(G,H)$ to be the least integer $r\geq0$ such that every finite genuine $G$-spectrum satisfies

$$K(n+r)_*(\Phi^H X)=0 \quad\Longrightarrow\quad K(n)_*(\Phi^G X)=0.$$

The higher starting height compensates for the information that taking more fixed points can lose.

A subnormal chain with cyclic quotients is a sequence

$$H=G_0\triangleleft G_1\triangleleft\cdots\triangleleft G_s=G$$

such that each group is normal in the next and each quotient $G_i/G_{i-1}$ is cyclic. Let $\ell(G,H)$ be the minimum possible length $s$. Such chains exist in finite $p$-groups, even when $H$ is not normal in $G$. A cyclic quotient of order $p^a$ counts as one step, not $a$ steps.

3. Development history and previous partial results

Smith theory established the classical fixed-point acyclicity phenomenon. Floyd strengthened it to inequalities for total mod-$p$ Betti numbers. Chromatic versions replace a single homology theory by the hierarchy of Morava theories and ask for the precise change of height.

Balmer and Sanders classified the prime tensor ideals of finite genuine equivariant spectra, reducing the topology of their spectrum to inclusions indexed by subgroups and heights. The cyclic prime-order case already displayed a one-step chromatic loss.

Barthel, Hausmann, Naumann, Nikolaus, Noel, and Stapleton determined the finite abelian $p$-group case. There, the loss is the minimum number of generators of $G/H$. This corrected an earlier order-based expectation: a cyclic group of order $p^2$ has loss one, not two. Balderrama and Kuhn later supplied an elementary proof of the abelian theorem, including rational target height zero.

The nonabelian case is subtler because quotient ranks need not capture the placement of a subgroup. Kuhn and Lloyd proved that the relevant acyclicity implications are equivalent to inequalities between Morava-homology dimensions. They obtained the cyclic-chain answer for subgroup pairs in central extensions of elementary abelian 2-groups by a group of order two.

For example, a noncentral order-two subgroup in the dihedral group of order eight has loss two, although a basic elementary-abelian quotient bound gives only one. Kuhn and Lloyd proposed the general equality with cyclic length and separately asked about independence from height. M1 claims both, with the precise prior statements documented in its introduction and bibliography.

4. What the new manuscripts claim

M1, dated 24 September 2026, states

$$r_n(G,H)=\ell(G,H)$$

for every prime $p$, every finite $p$-group $G$, every subgroup $H$, and every $n\geq0$.

The assertion includes optimality. If $r<\ell(G,H)$, there exists a finite genuine $G$-spectrum with

$$K(n+r)_*(\Phi^H X)=0, \qquad K(n)_*(\Phi^G X)\neq0.$$

Thus the formula is not merely an upper bound obtained by iterating known cyclic results. It says that no smaller loss works uniformly over all finite inputs. When $H=G$, both sides are zero.

A concrete abelian example illustrates why the embedded subgroup matters. Take

$$G=C_{p^2}\times C_p, \quad H_1=pC_{p^2}\times0, \quad H_2=0\times C_p.$$

Both subgroups are cyclic of order $p$, but

$$G/H_1\cong C_p\times C_p, \qquad G/H_2\cong C_{p^2}.$$

The claimed losses are two and one, respectively. Knowing only the abstract isomorphism type of $H$ is insufficient.

Combined with the existing conjugacy and $p$-subnormal reductions, the manuscript also gives the remaining finite-height prime-inclusion tests for arbitrary finite groups. Together with the previously known infinite-height information, this determines their thick tensor ideals. The general finite-group conclusion uses those reductions; it does not apply the displayed $p$-group formula without first checking the required subgroup conditions.

5. Why the advance matters

Importance: 5/5, provisionally. If correct, the theorem supplies the exact numerical invariant for a central fixed-point problem and completes the corresponding structural description of finite equivariant stable homotopy.

Its appeal is the match between two very different measurements. On one side are vanishing tests in generalized homology, quantified over every finite equivariant spectrum. On the other is the shortest length of a group-theoretic chain. The equality makes the former accessible through the latter.

Height independence is an additional structural statement. Increasing chromatic complexity does not change the amount of loss attached to a fixed subgroup pair. The same group invariant governs rational detection at height zero and all positive target heights.

6. The main idea

The upper bound and the missing converse

The upper bound follows by applying the cyclic chromatic Smith theorem at each step of a shortest chain. Each cyclic quotient costs at most one height, so the total cost is at most $\ell(G,H)$.

Sharpness is harder. One needs examples preventing every smaller proposed bound. Standard generalized character theory is indexed by maps from products of $p$-adic integers, whose images are abelian. That cannot directly see all the nonabelian subgroup configurations responsible for the problem.

Build a source with enough nonabelian information

The manuscript constructs a pro-$p$ source group from finite groups detected by telescopic cochains. A pro-$p$ group is an inverse limit of finite $p$-groups, retaining compatible information from all of them. The construction bounds the number of generators of every finite section of this source.

Relative integration over finite groupoids then imposes divisibility conditions on its finite actions. A groupoid allows objects with automorphisms; integration weights those automorphisms in a controlled way. Applying it to labeled finite sets produces congruences for fixed-point counts at every open subgroup.

The claimed profinite argument combines these arithmetic congruences with the generator bound to force a subnormal series with $\mathbb Z_p$ factors of bounded length. Thus a chromatic detection statement imposes a cyclic-length constraint on finite quotients. The paper emphasizes that constraints at all open subgroups, rather than only at the original group, carry essential information.

Turn a vanishing estimate into sharp finite examples

The first estimate is rational. Nilpotence and generalized characters upgrade it to higher heights. Products with increasingly large cyclic $p$-groups separate the target chromatic height from the group’s cyclic length.

A final support argument handles an arbitrary embedded subgroup. The support of a finite spectrum records the prime tensor ideals that do not contain it. At a hypothetical first failure of the desired bound, the new estimates would separate the support into two disjoint nonempty closed pieces. But that support is the closure of one point and must be irreducible, so such a separation is impossible.

This supplies the necessary prime-inclusion obstruction. Separation of primes then yields the finite spectra witnessing sharpness. The manuscript does not need to write down a small explicit cell model for every subgroup pair.

Telescopic localization is used in these intermediate estimates. On finite spectra its acyclicity tests agree with Morava $K$-theory, but the argument does not identify the two localizations on arbitrary spectra. Keeping that distinction avoids importing an unrelated telescope-conjecture assumption.

7. Scope, limitations, and open questions

The theorem quantifies over finite genuine spectra and their geometric fixed points. It should not be transferred automatically to arbitrary infinite spectra, naive group actions, or homotopy fixed points.

The value is relative to the actual inclusion $H\leq G$, not merely the orders or abstract types of the two groups. A chain may have quotients of different cyclic orders, and each contributes one unit.

The existence of sharpness witnesses is structural. It does not provide a uniform efficient algorithm for constructing a smallest witness or a bound on its number of cells. Likewise, a classification of tensor ideals does not compute every equivariant homotopy group.

For general finite groups, conjugacy and the appropriate common $p$-residual conditions remain part of the prime-inclusion criterion. The $p$-group theorem is the numerical component of that larger description, not a substitute for its hypotheses.

8. Sources and further reading

M1 contains the equality, its historical development, and the character, profinite, and support arguments. Its bibliography identifies Balmer–Sanders, The spectrum of the equivariant stable homotopy category of a finite group (2017); Barthel and collaborators, The Balmer spectrum of the equivariant homotopy category of a finite abelian group (2019); and Kuhn–Lloyd, Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups (2024), the principal predecessors of the stated classification. The Hopkins–Smith nilpotence theorem supplies the underlying finite-spectrum chromatic framework.

Source manuscripts

• M1. OpenAI, Cyclic length and chromatic fixed-point loss (September 24, 2026).

Sources consulted on 22 October 2026 at repository revision 1581d4c. Download the manuscript citations.